Visual attention guides our gaze to relevant parts of the viewed scene, yet the moment-to-moment relocation of gaze can be different among observers even though the same locations are taken into account. Surprisingly, the variability of eye movements has been so far overlooked by the great majority of computational models of visual attention. In this paper we present the ecological sampling model, a stochastic model of eye guidance explaining such variability. The gaze shift mechanism is conceived as an active random sampling that the foraging eye carries out upon the visual landscape, under the constraints set by the observable features and the global complexity of the landscape. By drawing on results reported in the foraging literature, the actual gaze relocation is eventually driven by a stochastic differential equation whose noise source is sampled from a mixture of α-stable distributions. This way, the sampling strategy proposed here allows to mimic a fundamental property of the eye guidance mechanism: where we choose to look next at any given moment in time, it is not completely deterministic, but neither is it completely random To show that the model yields gaze shift motor behaviors that exhibit statistics similar to those displayed by human observers, we compare simulation outputs with those obtained from eye-tracked subjects while viewing complex dynamic scenes.